A Government spokesperson said: "With plenty of discussions still to go, Monday will be an important staging post on the road to the crucial December Council."

That meeting, in 10 days' time, is when the Prime Minister hopes the remaining 27 EU countries will agree to end the deadlock and allow trade talks to begin.

Image:Michel Barnier, the EU chief Brexit negotiator, will also meet the PM and David Davis

After the Cabinet backed a greatly increased offer, up to around £40bn, she will be confident of winning the backing of Mr Juncker and Mr Barnier on the "divorce bill".

But the Government faces a tricky task in agreeing a deal on the Irish border that is acceptable to the EU, the Dublin government and Northern Ireland's Democratic Unionist Party.

Downing Street has revealed that the UK government and EU negotiators met throughout the weekend to try to reach agreement on the issues which are yet to be resolved.

And the PM received a boost ahead of the Brussels talks when Ireland's new deputy prime minister, Simon Coveney, said his country did not want to veto progress in Brexit negotiations.

"We certainly don't want to be vetoing anything," he said. "The Irish government, just like the British government, wants to be able to move the Brexit process on to phase two.

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Video:Donald Tusk: Key to UK's Brexit future lies in Dublin

"We want to be able to provide the kind of certainty that many businesses are calling for in Britain and Ireland, and indeed in other parts of the European Union.

"So there is no desire I can tell you in Ireland to delay this process."

Also, a source close to the negotiations on Ireland has told Sky News: "There's been some progress on the wording from both sides, but there's a bit of a distance to go.

"It's probably about 50/50 whether there will be agreement in the near future."

After their lunch with Mr Juncker and Mr Barnier, the PM and Mr Davis will brief Donald Tusk, the European Council president, who has to recommend to the 27 EU countries at the 14-15 December summit that trade talks can begin.

Up to now, Mr Tusk has said they cannot begin unless the UK can satisfy Dublin that there will be no return to a "hard border" between Northern Ireland and the Republic.

But as well as tough negotiations in Brussels with Mr Juncker, Mr Barnier and Mr Tusk, the Prime Minister is facing pressure from hardline Tory Brexiteers at Westminster, who want her to threaten to walk away.

Tories including Owen Paterson, Jacob Rees-Mogg, John Redwood and former chancellor Lord Lawson have signed a letter calling on her to refuse to settle the "divorce bill" unless Brussels agrees to new demands.

They include settling the terms of a free trade agreement "in principle", an end to the jurisdiction of the European Court of Justice and freedom of movement to the UK for EU nationals after Brexit.